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cofinite submodule; $\oplus $-cofinitely supplemented module
Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $ \oplus $-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus $-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus $-cofinitely supplemented modules is $\oplus $-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus $-cofinitely supplemented. In addition, if $M$ has the summand sum property, then $M$ is $\oplus $-cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of $M$.
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